Critical sets, crowns and local maximum independent sets

A set S⊆ V(G) is independent if no two vertices from S are adjacent, and by Ind (G) we mean the set of all independent sets of G. A set A∈ Ind (G) is critical (and we write A∈ CritIndep(G)) if | A| - | N(A) | = max { | I| - | N(I) | : I∈ Ind (G) } [37], where N(I) denotes the neighborhood of I. If S∈ Ind (G) and there is a matching from N(S) into S, then S is a crown [1], and we write S∈ Crown(G). Let Ψ (G) be the family of all local maximum independent sets of graph G, i.e., S∈ Ψ (G) if S is a maximum independent set in the subgraph induced by S∪ N(S) [22]. In this paper, we present some classes of graphs where the families CritIndep(G), Crown(G), and Ψ (G) coincide and form greedoids or even more general set systems that we call augmentoids.